| Chapter 1. GrSbner Bases, an Introduction Arjeh M. Cohen 1. Introduction 2. Monomials 3. The Buchberger Algorithm 4. Standard Monomials 5. Solving Polynomial Equations 6. Effectiveness of Polynomial Rings Chapter 2. Symbolic Recipes for Polynomial System Solving Laureano Gonzalez-Vega, Fabrice Rouillier,and Marie-Frangoise Roy 1. Introduction 2. General Systems of Equations 2.1 Algebraic Preliminaries 2.2 First Recipes for Polynomial System Solving 3. Linear Algebra, Traces, and Polynomial Systems 3.1 Eigenvalues and Polynomial Systems 3.2 Counting Solutions and Removing Multiplicities 3.3 Rational Univariate Representation 4. As Many Equations as Variables 4.1 Generalities on Complete Intersection Polynomial Systems 4.2 Recipes for Polynomial System Solving When the Number of Equations Equals the Number of Unknowns 5. GrSbner Bases and Numerical Approximations Chapter 3. Lattice Reduction Frits Beukers 1. Introduction 2. Lattices 3. Lattice Reduction in Dimension 2 4. Lattice Reduction in Any Dimension 5. Implementations of the LLL-Algorithm 6. Small Linear Forms Chapter 4. Factorisation of Polynomials Frits Beukers 1. Introduction 2. Berlekamp's Algorithm 3. Additional Algorithms 4. Polynomials with Integer Coefficients 5. Factorisation of Polynomials with Integer Coefficients, I 6. Factorisation of Polynomials with Integer Coefficients, II 7. Factorisation in K[X], K Algebraic Number Field Chapter 5. Computations in Associative and Lie Algebras Ggbor Ivanyos and Lajos Rdnyai 1. Introduction 2. Basic Definitions and Structure Theorems 3. Computing the Radical 4. Applications to Lie Algebras 5. Finding the Simple Components of Semisimple Algebras 6. Zero Divisors in Finite Algebras Chapter 6. Symbolic Recipes for Real Solutions Laureano Gonzalez-Vega, Fabrice Rouillier, Marie-Frangoise Roy,and Guadalupe Trujillo 1. Introduction 2. Real Root Counting: The Univariate Case. 2.1 Computing the Number of Real Roots 2.2 Sylvester Sequence 2.3 Sylvester-Habicht Sequence 2.4 Some Recipes for Counting Real Roots 3. Real Root Counting: The Multivariate Case 4. Tile Sign Determination Scheme 5. Real Algebraic Numbers and Thorn Codes 6. Quantifier Elimination 7. Appendix: Properties of the Polynomials in the Sylvester-Habicht Sequence 7.1 Definition and the Structure Theorem 7.2 Proof of the Structure Theorem 7.3 Sylvester-Habicht Sequences and Cauchy Index Chapter 7. GrSbner Bases and Integer Programming Giinter M. Ziegler 1. Introduction 2. What is Integer Programming? 3. A Buchberger Algorithm for Integer Programming …… Chapter 8. Working with Finite Groups Chapter 9. Symbolic Analysis of Differential Equations Chapter 10. Grobner Bases for Codes Chapter 11. Grobner Bases for Decoding Project 1. Automatic Geometry Theorem Proving Project 2. The Birkhoff INterpolation Problem Project 3. The Inverse Kinematics Problem in Robotics Project 4. Quaternion Algebras Project 5. Explorations with the Icosahedral Group Project 6. The Small Mathieu Groups Project 7. The Golay Codes Index |
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